Gaussian Elimination (Row Reduction)
Gaussian elimination is an algorithm for solving systems of linear equation. It is avaible when the number of equations equals or more than the number of unknowns. For example;
1x + 3y + 1z = 9
x + y – z = 1
y +2z = 35
There are 3 unknowns and 3 equations so, we can apply gaussian elimination to reach the set of solution. Let’s see..
First we have to write the equation system as a matrix. Then continue row operations until obtain a matrix form like the given below.
This form is also called row echelon form. Also it is called reduced row echelon form, if there are zeros instead of * (stars). After obtain the row echelon form, we can reach the set of solution.
The only difference between Gaussian Eliminaton and Gauss-Jordan Elimination is continue to row reduction operations until obtain reduced row echelon form instead of row echelon form so, we reach the set of solution directly.
Inverse of a Matrix
There are different ways to find the inverse of a matrix. The first way is finding inverse of a matrix with elementary row operations. Write the given matrix with an identity matrix as an augmented matrix like the given below;
The given matrix is on left side and the identity matrix on right side. We are continue to elementary row operations until obtain an identity matrix on the left side. When we obtain an identity matrix on the left side, we obtain inverse of the given matrix on the right side. Thats all.
Also we can find inverse of a given matrix with the formula given below;
As seen at the formula, inverse of a matrix A is equal to multiplication of adjacency matrix of A with 1 over determinant of matrix A. Also adjacency matrix is equals to transpose of coefficient matrix of A.